sequence and series

Question

For real number α, if ∑^∝ a_n and ∑^∝ b_n converges to A andB respectively , then ∑^∝ ( a_n + α b_n) 

                              ^{n=1               n=1                                                                  n=1}

 

a) converge to A+ αB 

b) convergeto αA + B

c) diverge

                                                                      

 

Answer 1

There is a rule for breaking series up:

∑(a_n + b_n) = ∑a_n + ∑b_n.

 

There is also a rule for multiplying each term in a series by a constant, c:

∑ca_n = c∑a_n.

 

These rules apply also to convergent infinite series.  Since writing the limits for sums here is a messy business, assume their existence below (for n=1 to ∞).  Applying the first and second rules, in turn, yields

 

∑(a_n + αb_n) = ∑a_n + ∑αb_n

=∑a_n + α∑b_n

=A + αB (which is, again, convergent.)